Archimedes' Method for Computing Areas and Volumes - Exercise on Proposition 6 of The Method

Archimedes' Proposition 6, illustrated below, concerns a cone inscribed in a hemisphere. If you click on Proposition 6, you will see a dynamic version of the same sketch.

a) Describe the balancing relationship that is illustrated in the sketch.

b) Use this to determine the location of the center of gravity of the hemisphere. You will need to use the fact that the center of gravity of a cone lies on the cone's axis, three times as far from the vertex as from the base.